Emissions from the OH Meinel bands are routinely used to determine rotational
temperatures that are considered proxies for the kinetic temperature near the
mesopause region. Previous observations determined OH rotational temperatures
that show a dependence on the vibrational level, with the temperature rising
overall as the OH vibrational quantum number v increases. The source of
this trend is not well understood and has generally been attributed to
deviations from thermodynamic equilibrium. This technical note demonstrates
that the existence of bimodal OH rotational population distributions is an
inherent feature of rotational relaxation in gases and can provide an
explanation for the previously reported temperature trend. The use of only a
few lines from rotational transitions involving low rotational quantum
numbers to determine rotational temperatures does not account for the
bimodality of the OH rotational population distributions and leads to
systematic errors overestimating the OH rotational temperature. This note
presents selected examples, discusses the relevant implications, and
considers strategies that could lead to more reliable OH rotational
temperature determination.

The hydroxyl radical is an important species in the middle atmosphere of the
Earth. At altitudes of around 87 km, the exothermic reaction of ozone with
atomic hydrogen produces rotationally and vibrationally excited hydroxyl,
OH(v), in vibrational levels v=5–9 (Adler-Golden, 1997; Khomich et al.,
2008; von Savigny, 2017; and references therein). The radiative decay of
OH(v) in the visible and infrared regions of the electromagnetic spectrum,
known as the OH Meinel band emission, is a prominent feature in night sky
spectra. The OH Meinel band emission has been used to monitor atmospheric
density changes, temperature fluctuations, and species concentrations for
several decades (Meriwether, 1989; Sivjee, 1992; Khomich et al., 2008;
Grygalashvyly, 2015).

Collisional relaxation of OH(v) by other atmospheric species plays an
important role in determining the observed internal quantum-state
distribution. As a result, collisional energy transfer between OH(v) and the
major components of the atmosphere at this altitude region, O2 and
N2, have been studied for many years. Nevertheless, several gaps
persist in our knowledge of these processes. Especially for oxygen atoms,
which form a significant component of the atmosphere at the high-altitude
part of the OH(v) layer, studies of collisional energy transfer have been
relatively limited. Notable recent developments from laboratory studies
include the demonstration that the deactivation of OH(v=9) by O atoms is
characterized by a total loss rate coefficient that is significantly larger
than that by O2 and N2, and the most efficient relaxation pathway
involves multi-quantum vibrational-to-electronic energy transfer
(Kalogerakis et al., 2011, 2016).

The question of whether the OH rotational temperature determined by
observations is equivalent to the local kinetic temperature is of fundamental
significance and has been debated since the discovery of the Meinel band
emission in the 1950s (Kalogerakis et al., 2018, and references therein).
Simultaneous observations of mesospheric OH(v) emissions from several
vibrational levels by Cosby and Slanger (2007) and Noll et al. (2015, 2016)
using sky spectra from astronomical telescopes reported rotational
temperatures that exhibit a clear vibrational level dependence; the
rotational temperature increases by approximately 15 K as the OH vibrational
quantum number increases from v=2 to v=8. Both groups also determined
that the rotational temperature of OH(v=8) was significantly higher than
that for OH(v=9). Figure 1 summarizes the results on OH rotational
temperatures reported by Cosby and Slanger (2007), Oliva et al. (2015), and
Noll et al. (2016). Despite some variation, possibly due to the different
location, time, and instrument for these measurements, these data sets show a
similar trend for the vibrational level dependence of the OH rotational
temperatures. Additional details and the Oliva et al. (2015) data set can be
found in the original publications as well as in Kalogerakis (2017).

In this technical note, we first briefly consider the available knowledge
from fundamental theoretical and experimental studies of rotational energy
transfer. These studies unambiguously demonstrate that bimodal rotational
population distributions are an inherent feature of the rotational
relaxation process in gases. Signatures of bimodal behavior have been
observed in the laboratory as well as in the upper atmosphere. We then show
that neglecting to account for this bimodality in the mesospheric OH
rotational population distributions leads to large systematic errors in the
determined rotational temperatures. These findings provide an explanation
for the aforementioned dependence of the OH rotational temperatures on the
vibrational level determined in previous studies. Finally, this note briefly
discusses the implications for mesospheric temperature measurements and
strategies for mitigation of systematic errors.

Before considering results from atmospheric observations, it is highly
informative to review selected information from theoretical studies on the
mechanism of rotational relaxation as well as relevant laboratory results.

In their seminal experiments on rotational energy transfer investigated with
the technique of “arrested relaxation” using infrared chemiluminescence,
John C. Polanyi and coworkers (Charters and Polanyi, 1962; Anlauf et al.,
1967; Polanyi and Woodall, 1972) investigated how the initial highly
rotationally excited nonthermal population distribution of hydrogen chloride
from the H+Cl2 reaction attained thermal equilibrium in collisions
with the bath gas. A general observation in these studies was that
rotational energy transfer was less efficient as the rotational excitation
increased or, in other words, as the energy spacing between rotational
levels became larger. A key finding by Polanyi and coworkers was that
rotational-to-translational (R–T) energy transfer of an initial rotationally
excited population distribution peaking at high rotational quantum number
J does not exhibit a transient peak at intermediate J values. Instead, a
bimodal distribution is generated with a peak at high J, reflecting the
nascent excited rotational population distribution, as well as a new
secondary peak at low J, corresponding to the thermal distribution of the
bath gas. As the rotational relaxation process progresses, the amplitude of
the excited nonthermal population distribution decreases while that of the
thermalized distribution increases accordingly. Polanyi and Woodall (1972)
developed a theoretical model for R–T energy transfer that quantitatively
accounted for their experimental observations. According to this model, the
transition probability for rotational energy transfer decreases
exponentially with the energy gap between the two rotational states involved
in the rotational energy exchange. Figure 2 demonstrates the bimodal pattern
observed when an initial rotational population distribution that is a delta
function relaxes according to the model of Polanyi and Woodall (1972).
Exponential-gap models have been extensively used in studies of rotational
energy transfer for decades (Koszykowski et al., 1985; Lucht et al., 1986;
Dodd et al., 1994; Holtzclaw et al., 1997; Beaud et al. 1998; Fei et al.,
1998; Kliner and Farrow, 1999; Hickson et al., 2002; Knopp et al., 2003;
Funke et al., 2012; Noll et al., 2018).

Figure 2Temporal evolution of a delta function initial rotational
population distribution relaxing in a bath gas according to an
exponential-gap model with unrestricted ΔJ. Adapted from Fig. 4 of
Polanyi and Woodall (1972). The alternating black and grey labels indicate
reduced time units.

Regarding rotational relaxation involving the hydroxyl radical, Kliner and
Farrow (1999) performed relevant laser-based experiments studying energy
transfer in OH(v=0) excited to rotational levels N=1–12 near room
temperature. In those studies, pulsed photolysis of H2O2 at 266 nm
created a rotationally excited population distribution, whose temporal
evolution was probed using laser-induced fluorescence (LIF). Kliner and
Farrow (1999) were able to determine that rotational relaxation by O2
and N2 is more efficient for lower rotational levels than for higher
ones. They also found that an exponential gap model similar to that of
Polanyi and Woodall (1972) reproduced their laboratory measurements
remarkably well. Figure 3 presents the results of Kliner and Farrow (1999)
for rotationally excited OH(v=0, N=1–12) colliding with N2 bath
gas. The figure also shows Boltzmann fits to the data of Kliner and Farrow (1999)
using a fitting function described by two Boltzmann distributions, at
low and high temperatures. The initially excited and the final thermalized
distributions were fit to a single-temperature, indicated in Fig. 3. For the
other measurements, we constrained the two determined temperatures as fixed
values and varied the partitioning of the two rotational level populations
at the two characteristic temperatures so as to reflect the changes in the
degree of thermal equilibration. As Fig. 3 shows, this experimental system
is well described by a low temperature value near room temperature and a
high temperature value (293±4 and 1567±38 K, based on our
fits) reflecting the nascent rotational distribution of OH(v=0)
following photodissociation of H2O2. Gericke et al. (1986)
performed a relevant laboratory study investigating the dynamics of
H2O2 photodissociation at 266 nm and found that the nascent OH(v=0) product rotational state distribution was characterized by a
temperature of 1530±150 K, in excellent agreement with the results
of the fits shown in Fig. 3. We find similar agreement with the measurements
of Kliner and Farrow (1999) for collider gases O2 and Ar. In summary,
the results of Kliner and Farrow (1999) provide further validation for the
exponential-gap rotational relaxation model of Polanyi and Woodall (1972) as
well as a clear laboratory demonstration of bimodality in the OH product
state distributions following rotational relaxation.

Figure 3Experimental results (circles) of Kliner and Farrow (1999) for
rotational relaxation of OH(v=0, N=1–12) colliding with N2 bath
gas and fits to single-temperature and two-temperature Boltzmann
distribution functions. The two characteristic temperatures represent the
initial OH distributions and the final fully thermalized gas. The relative
weight of the two Boltzmann distributions changes as the relaxation process
evolves in time. The grey labels show the product time × pressure
corresponding to the experimental measurements.

Atmospheric observations have revealed that the rotational population
distributions of mesospheric OH display a bimodal character. Early
observations provided the first indications for emission from lines
associated with high rotational excitation in selected vibrational levels
(Pendleton Jr. et al., 1989, 1993; Perminov and Semenov, 1992; Dodd et al., 1993; Perminov et al., 2007; and references therein).
Recent simultaneous observations of multiple OH vibrational levels using
high-resolution spectrographs from astronomical telescopes by Cosby and
Slanger (2007) and Oliva et al. (2015) represent the most comprehensive
demonstrations of bimodal behavior in OH(v) rotational population
distributions to date. Bimodal behavior is evident for all observed
vibrational levels OH(v=2–9), but this effect may appear at first less
pronounced for the highest vibrational levels. In fact, the opposite is true
because the higher the OH vibrational level is, the larger the fraction of
the rotational level population that deviates from thermodynamic
equilibrium. This behavior results from the fact that the OH(v) radiative
lifetime decreases as the vibrational level increases (Brooke et al., 2016)
and, consequently, the higher OH vibrational levels experience fewer
collisions with the ambient atmosphere. In principle, more complex behavior
than bimodal might be possible because of the large number of production and
removal pathways for mesospheric OH(v). Hints of additional features may be
discerned for v=3–5 in the data of Oliva et al. (2015), but these are at
best tentative given the signal-to-noise ratio. Additional measurements at
high resolution and sensitivity combined with careful corrections for any
absorption and spectral interferences will be required to settle this
question. Based on the available information to date, it appears that to a
first approximation the simplest adequate description of the mesospheric
OH(v) rotational population distributions is that of bimodal Boltzmann
distributions.

We now consider the effect of bimodal OH rotational population distributions
on the determination of OH rotational temperatures by considering an example
for OH(v=9). This is the highest populated vibrational level and most
probable product of the H+O3 reaction. Collisional cascade from
higher vibrational levels can be assumed to be a limited, and most likely
negligible, source. Therefore, more than any other OH vibrational level,
rotational relaxation of v=9 is expected to follow the exponential-gap
model of Polanyi and Woodall (1972). Figure 4 presents the observed
rotational population distribution reported by Noll et al. (2018) for v=9
together with fits we performed using one simple and one bimodal Boltzmann
distribution function. In the former case, only rotational lines with energy
less than 250 cm−1 are considered. From Fig. 4, we conclude that
neglecting the bimodal behavior of the rotational population distributions
and considering only a few rotational lines involving the lowest quantum
numbers leads to unacceptably large systematic errors in the extracted OH
rotational temperatures. The lower temperature value for the bulk of the
population obtained from the fit using a two-temperature bimodal Boltzmann
distribution is 20 K lower than the temperature obtained using a single
Boltzmann function and only a few low-level rotational transitions.

Figure 4Mesospheric OH(v=9) rotational population distribution based
on the observations of Noll et al. (2018; Fig. 3b). The grey dotted line
shows the result of a single-temperature fit for E< 250 cm−1.
The black solid line shows a two-temperature fit using all the data points.
Not considering the bimodality of the rotational population results in large
systematic errors because the contributions of the non-thermalized Boltzmann
distribution to the low rotational energy region are not accounted for.

We recently considered two-temperature fits for selected OH vibrational
bands from the Oliva et al. (2015) data set (Kalogerakis et al., 2018). The
OH(v) rotational temperatures inferred from single and two-temperature
Boltzmann distribution functions are generally different. Based on the
information above, it becomes clear the observed trend for
single-temperature fits does not reflect real temperature changes; it is
an artifact that arises from neglecting the bimodal character of the OH rotational population distribution.
Although this effect is most pronounced for the largest OH(v) levels, e.g.,
v=8, 9, differences for the lowest observed vibrational levels, v=2,
3, appear to be comparable to the estimated uncertainties. The majority of
the OH(v) product from the H+O3 reaction is generated in the highly
vibrationally excited levels v=7–9, while collisional or radiative
relaxation is needed to generate the lowest vibrational levels. Thus, it is
reasonable to expect that the lower vibrational levels have undergone more
extensive thermalization. At the same time, however, we do not fully
understand all the relevant collisional relaxation processes and the
variability of the bimodal character in the OH rotational population
distributions. Thus, although we could state with confidence that not
accounting for the bimodality in the rotational population distributions
introduces large systematic errors for the highest OH vibrational levels, it
is presently difficult to assess the extent to which changes in the
rotational temperatures for OH(low-v) are influenced by variations in the
fraction of the rotational population distribution that is not in thermal
equilibrium.

The most important finding of this technical note is the demonstration that
the traditional approach in aeronomy to determine OH rotational temperatures
using only a pair or a few rotational lines involving the lowest rotational
quantum levels does not account for the bimodality of the observed
mesospheric OH rotational population distributions and can lead to
unacceptably large systematic errors in the OH rotational temperature
determination, especially for OH(high v). To mitigate this problem, the
recommended approach would be to concurrently obtain information on the
non-equilibrated, high-rotational-level tail of the OH(v) rotational
population distribution. The adequate resolution and sensitivity to record
the full rotational population distribution may not always be available, but
even establishing a lower limit for the ratio of the high-rotational-level
population versus the low-rotational-level population would be helpful in
assessing potential systematic errors. Without this type of information, it
is not clear what portion of the observed variability in the OH rotational
temperature of any specific vibrational level could be attributed to changes
in the non-thermalized rotational population.

Evidence from theoretical calculations and laboratory experiments
demonstrates that rotational energy transfer between a Boltzmann
distribution of rotationally excited molecules and a thermal bath leads to
bimodal distributions. Such behavior has indeed been reported in
atmospheric observations of mesospheric OH. The common approach in aeronomy
of considering only a few OH rotational lines with the lowest rotational
excitation to determine the rotational temperature does not account for the
bimodality of the OH(v) rotational population distributions and can lead to
large systematic errors overestimating the rotational temperature. These
errors are largest for the highest OH(v) vibrational levels and their
magnitude can reach several degrees Kelvin. This effect provides an
explanation for the apparent vibrational-level dependence of OH rotational
temperatures reported from previous atmospheric observations. Careful
consideration of the highly rotationally excited portion of the rotational
population distributions under study is required for a reliable
determination of rotational temperatures from mesospheric OH(v) Meinel band
observations.

The data set from Oliva et al. (2015) presented here and
information relevant to the analysis are available on the Open Science
Framework website https://doi.org/10.17605/OSF.IO/NKWPJ (Kalogerakis, 2017).

This article is part of the special issue “Layered phenomena in the mesopause region (ACP/AMT inter-journal SI)”.
It is a result of the LPMR workshop 2017 (LPMR-2017), Kühlungsborn, Germany, 18–22 September 2017.

The material presented here is based in part on work supported by the US
National Science Foundation (NSF) grant AGS-1441896, NASA grant
80NSSC17K0638, and SRI International research and development funds. The author thanks Philip C. Cosby,
Tom G. Slanger, and Stefan Noll for helpful discussions.

Light emission from energetic hydroxyl radical, OH*, is a prominent feature in spectra of the night sky. It is routinely used to determine the temperature of the atmosphere near 90 km. This note shows that the common practice of using only a few emission features from low rotational excitation to determine rotational temperatures does not account for the bimodality of the OH population distributions and can lead to large systematic errors.

Light emission from energetic hydroxyl radical, OH*, is a prominent feature in spectra of the...